By H.; Beckert, H. Schumann

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**Example text**

22) in order to arrive at 9, and 9' within φ3 is added to 9 in φι to produce 9'. Therefore: 36α2+68α-1 16 ' ,,24α2+40α-1 16 2 (l + 2af _ Q5a 16 +9a 2 4α2+16«-1_ 8 „,32α 2 + 7 2 α - 3 16 For further information on the meanings of the 9 and 9' terms it is best to consult von Mangoldt's Mathematische Annalen paper (1905). An additional term is also placed within Eq. π 2u ,u, K- + a2(3a + l)T2 1 \τ-β\ In this expression K and β are associated with complex numbers. Eq. 26). 26) At this point the analysis of the integration procedure is paused owing to the complicated nature of the emerging error term.

In other words it is possibly best to start the integration from Eq. 12) instead of Eq. 11). 35 The Functional Equation Sec. 2] δ->ο ^ +00 X +00 s. 2: A possible alternative path for arrival at the functional equation by line integration. 1 integral j - -dx. 0. 020 36 The Riemann Zeta Function [Ch. 2 x ? 0 'S? 2 ? 3: Variation of — - — with increasing x at different σ values. 1. (e*- )x Sec. 4: Contour used by Elizalde and Brevik during a study of Casimir energy theory. 4 Connections to quantum field theory Zeta function regularization is a technique that can be used to derive Casimir energies associated with vacuum energies present between atoms.

_ T T N(T) = — In 2π 5 2π a 13 Ζ + η +— + 2π 16 1 + 4α (5 - +— 8 2 AT ---L9. 4) Eq. 5) is representative of N(T) when σ = 1. 5) On returning to the derivation of (|>i(a,T) or φ^Τ-ia) that spans from Eq. 5) to T A l + 2a Eq. 21), the important stage to identify is multiplication of /'—by — 2T , in Eq. 19). —L Γ i_±_^. j In - Eq. 19) originates from In [ - + - + /'— j or in Eq. 18). 19) produces in Eq. 20), which von Mangoldt surprisingly writes as in Eqs. 22). 1 can be produced. Using this simplified procedure meant that the important θ[ terms could be quickly identified.

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